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Chapter 1 - Hyperplane arrangements

from Part I - Species and operads

Published online by Cambridge University Press:  28 February 2020

Marcelo Aguiar  and
Swapneel Mahajan
Marcelo Aguiar
Affiliation:
Cornell University, Ithaca
Swapneel Mahajan
Affiliation:
Indian Institute of Technology, Mumbai
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Summary

This chapter reviews important notions and results on hyperplane arrangements required in the book. The discussion includes: 1) geometric objects such as faces, flats, bifaces, partial-flats, nested faces, lunes, bilunes, cones; 2) algebraic objects such as the Tits monoid, Birkhoff monoid, Janus monoid and their linearized algebras; Lie and Zie elements; 3) combinatorial objects such as distance functions and Varchenko matrices; descent, lune, Witt identities; incidence algebras, zeta and Möbius functions along with their noncommutative and two-sided analogues, the Zaslavsky formula and its noncommutative analogue.

Keywords

faceflatluneTits productlune-incidence algebrabilune-incidence algebranoncommutative zeta and Möbius functionsnoncommutative Zaslavsky formulaTakeuchi elementLie elementZie elementdistance functionVarchenko matrixlune identitydescent identityWitt identity
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Bimonoids for Hyperplane Arrangements , pp. 19 - 72
Publisher: Cambridge University Press
Print publication year: 2020

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